A linearized Riemann solver uses a linearization of the vector flux
in
Eq. (41)
around reference value
to bring it into a form similar to the scalar
Eq. (194),

(203)

Dropping the 2nd order term and
choosing
we get

(204)

(205)

We remember from Sect. 1.5.3 that for the Jacobian

(206)

there is a matrix
that brings it into diagonal form